Time-frequency distributions are widely used more and more for non-stationary signal analysis. They perform a mapping of one-dimensional signal x(t) into a two dimensional function of time and frequency TFDx(t,f) that yields a signature of the variation of the spectral content of the signal with time.
Many approaches are known in the art to perform the above-mentioned mapping. The most intuitive approach consists of analyzing the signal for small periods of time during which it can be assumed that the signal does not contain rapid changes. In the context of a slowly varying signal, this window concept will provide a useful indication of the variations over time.
The well-known spectrogram and the short-time Fourier transform are techniques that utilize the above window concept, and have become standard techniques in the art. These known systems, however, are not useful in situations where the energy, or spectral content of the signal, varies with such rapidity that the signal cannot reasonably be considered to be stationary for almost any window duration. In this regard, it is to be noted that, as the duration of the window is decreased, the frequency resolution of the system is also decreased.
As indicated, the spectrogram applies the Fourier transform for a short-time analysis window, within which it is assumed that the signal behaves reasonably within the requirements of stationarity. Moving the analysis window in time along the signal, one hopes to track the variations of the signal spectrum as a function of time. If the analysis window is made short enough to capture rapid changes in the signal, it becomes impossible to resolve frequency components that are close in frequency during the analysis window duration.
The well-known Wigner-Ville distribution provides a high-resolution representation in time and in frequency for a non-stationary signal, such as a chirp. However, it suffers from significant disadvantages. For example, its energy distribution is not non-negative and it is often characterized with severe cross terms, or interference terms, between components in different time-frequency regions. These cross terms lead to false manifestation of energy in the time frequency plan.
The Choi-Williams distribution allows reduction of such interferences compared to the Wigner-Ville distribution.
Since the spectrogram, Short-Time Fourier transform, Wigner-Ville and Choi-Williams distributions are believed to be well known in the art, they will not be described herein in further detail.
A general class of time-frequency distributions (TFD) is the Cohen's class distributions. A member of this class has the following expression:.
                                          TFD            x                    ⁡                      (                          t              ,              f                        )                          =                  ∫                      ∫                          ∫                                                ϕ                  ⁡                                      (                                          η                      ,                      τ                                        )                                                  ×                                  (                                                            t                      ′                                        +                                          τ                      2                                                        )                                ⁢                                                      x                    H                                    ⁡                                      (                                                                  t                        ′                                            -                                              τ                        2                                                              )                                                  ⁢                                  ⅇ                                                            -                      j2π                                        ⁢                                                                                  ⁢                    η                    ⁢                                                                                  ⁢                    t                                                  ⁢                                  ⅇ                                                            -                      j2π                                        ⁢                                                                                  ⁢                    τ                    ⁢                                                                                  ⁢                    f                                                  ⁢                                  ⅇ                                                            -                      j2π                                        ⁢                                                                                  ⁢                    η                    ⁢                                                                                  ⁢                                          t                      ′                                                                      ⁢                                  ⅆ                                      t                    ′                                                  ⁢                                  ⅆ                  τ                                ⁢                                  ⅆ                  η                                                                                        (        1        )            where t and f represent time and frequency, respectively, and H the transposed conjugate operator.
The kernel φ(η,t) characterizes the resulting TFD. It is known in the art that the use of a Cohen's class of distributions allows the definition of kernels whose main property is to reduce the interference patterns induced by the distribution itself.
An example of such a kernel is the Gaussian kernel that has been described in “KCS—New Kernel Family with Compact Support in Scale Space: Formulation and Impact”, from IEEE T-PAMI, 9(6), pp. 970–982, June 2000 by I. Remaki and M. Cheriet.
A problem with the Gaussian kernel is that it does not have the compact support analytical property, i.e. it does not vanish itself outside a given compact set. Hence, it does not recover the information loss that occurs due to truncating. Moreover, the prohibitive processing time, due to the mask's width, is increased to minimize the loss of accuracy.